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As stated above, Thales's theorem is a special case of the inscribed angle theorem (the proof of which is quite similar to the first proof of Thales's theorem given above): Given three points A, B and C on a circle with center O, the angle ∠ AOC is twice as large as the angle ∠ ABC. A related result to Thales's theorem is the following:
A special case of the theorem is Thales's theorem, which states that the angle subtended by a diameter is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle.
A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. [6] All such primitive triples can be written as (a, b, c) where a 2 + b 2 = c 2 and a, b, c are coprime. Thus a, b, c are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third ...
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.
The theorem can also be thought of as a special case of the intersecting chords theorem for a circle, since the converse of Thales' theorem ensures that the hypotenuse of the right angled triangle is the diameter of its circumcircle. [1]
Thales' theorem states that if is the diameter of a circle and is any other point on the circle, then is a right triangle with a right angle at . The converse states that the hypotenuse of a right triangle is the diameter of its circumcircle .
Diagram to show the converse to Thales' theorem. This is that given a right-angle, ABC, the circle centred on the midpoint of AC, O, passes through B. Date: 13 April 2007: Source: Own drawing, Inkscape 0.45: Author: Inductiveload: Permission (Reusing this file)
Proof of Apollonius' definition of a circle. First consider the point on the line segment between and , satisfying the ratio.By the definition | | | | = | | | | and from the converse of the angle bisector theorem, the angles = and = are equal.